Advances in Pure Mathematics, 2015, 5, 59-61 Published Online February 2015 in SciRes. http://www.scirp.org/journal/apm http://dx.doi.org/10.4236/apm.2015.52007

How to cite this paper: Mueckenheim, W. (2015) Sequences and Limits. Advances in Pure Mathematics, 5, 59-61. http://dx.doi.org/10.4236/apm.2015.52007

Sequences and Limits Wolfgang Mueckenheim University of Applied Sciences, Augsburg, Germany Email: wolfgang.mueckenheim@hs-augsburg.de Received 6 January 2015; accepted 23 January 2015; published 26 January 2015

Copyright © 2015 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract It is widely held that irrational numbers can be represented by infinite digit-sequences. We will show that this is not possible. A digit sequence is only an abbreviated notation for an infinite se-quence of rational partial sums. As limits of sequences, irrational numbers are incommensurable with any grid of decimal fractions.

Keywords Series, Sequences, Limits, Definability of Real Numbers, Set Theory

1. Introduction Strictly monotonic sequences do not assume their limit. Rarely the terms of the sequence and its limit are con-fused. But this situation changes dramatically when sequences of partial sums of series are involved. It is custo-mary in textbooks to identify the infinite sum over all terms of a series and the limit of this series [1], often called its “sum”. G. Cantor, one of the inventors of this habit, wrote: “ 3 ist also nur ein Zeichen für eine Zahl, welche erst noch gefunden werden soll, nicht aber deren Definition. Letztere wird jedoch in meiner Weise etwa durch (1.7, 1.73, 1.732, ...) befriedigend gegeben.” [2] 3 is only a symbol for a number which has yet to be found, but is not its definition. The number itself however is given satisfactorily in my way by (1.7, 1.73, 1.732, …).

In the following we will see that this is imprecise and point out an important consequence. A limit is not de-fined by the infinite sequence of partial sums because the sequence cannot be given in the necessary complete-ness. Only a finite formula can determine both the terms of the sequence of partial sums and the limit as well.

2. Theorem and Proof Theorem A non-terminating series of decimal fractions does not determine a real number.

Corollary A non-terminating digit sequence does not determine a real number. Proof. The limit of a strictly monotonic sequence is not among its terms. Strictly monotonic sequences like

W. Mueckenheim

60

( )10 n

n

−

∈ or [ ]( )1 1 n

nn

∈+

or !

110

nk

k n

−

= ∈

∑

sufficiently show this. None of the 0ℵ indexed terms is equal

to the limit 0, e, and Liouville’s number L, respectively. ( 0ℵ is the cardinality of the set of natural numbers n.) The same distinction has to be observed with series. There must not be a difference in the mathematical con-

tents whether the partial sums are written separately like 3, 3.1, 3.14, 3.141, 3.1415, (1)

or are written in one line with interruptions

( )( )( )( )( )3. 1 4 1 5 (2)

or without interruptions 3.1415 (3)

The infinite sequence of digits dn is completely exhausted by all terms of the Cauchy-sequence of rational partial sums of decimal fractions. The intended meaning as a sequence of rational partial sums according to Eq-

uation (1) can be expressed also by 0,1,2,

0 10nn

n

d=∑

. Equation (3) giving the infinite sum 310

nn

n

d∈

+ ∑

, is merely an

abbreviation: All partial sums are written in one and the same line without adding the limit. Equation (2) is the same because writing or not writing parentheses must not change the result. In all cases none of the 0ℵ decimal fractions is left out. The “sum” of the series, i.e., the limit of the Cauchy-sequence of partial sums, is not estab-lished by any term with natural index n∈ . But only all these 0ℵ terms are given in Equations (1) to (3) as well as on the left-hand sides of the following examples whereas the limits are given on the right-hand sides.

0

! ! !1 1

0 0

1 1

1 1 1lim ,10 10 10

1 1 1lim ,! ! !

1 1 1lim 1.2 2 2

n

n k knn k kn

nn k k

n

n k knn k k

L

en k k

∞

→∞∈ = =

∞

→∞∈ = =

∞

→∞∈ = =

≠ = =

≠ = =

≠ = =

∑ ∑ ∑

∑ ∑ ∑

∑ ∑ ∑

Digits are simply too coarse-grained to represent irrational limits of Cauchy-sequences.

To represent 1lim10nn→∞

by an infinite digit sequence, we would need infinitely many digits 0 preceding the

digit 1. Whereas it is obvious that this is impossible, the infinitely many digits 1 required for the expansion of

1 1

1 1 1lim910 10

n

k kn k k

∞

→∞ = =

= =∑ ∑ are usually swallowed without scruples. But it is as obvious that digits 0 and digits 1

do not allow for a different treatment with respect to the fact that never infinitely many can precede one of them. This leads us to the often asserted double-representation of periodic rational numbers. For all n∈ the sum

of the nth terms of the two complementary sequences

( )( )

( )

1 10 0.1,0.01,0.001, 0

1 1 10 0.9,0.99,0.999, 1

1 1 , 1 , 1 ,... 1

n

n

= →

− = →

= →

is 1. Since all 0ℵ digits are not sufficient to realize the limit 0 of the first sequence, all 0ℵ digits of 0.999… are not sufficient to realize the limit 1 of the second sequence. Only when explicitly taking the limits of the se-quences, we get 0 and 1, respectively. For series, taking the limit is usually assumed without saying and does not cause mistakes in numerical calculations, but if we look at the matter with advisable mathematical precision, we see

1 1

9 9 90.999 lim 110 10 10

n

n k nnn k n

∞

→∞∈ = =

= ≠ = =∑ ∑ ∑

W. Mueckenheim

61

The usual proof for 0.999 1= , namely 10 0.999 9.999 9 0.999 9 0.999 9⋅ = = + ⇒ ⋅ = holds in the limit only. The series 0.999… is not a number but a sequence of partial sums. Like a vector it can be multiplied such that ( ) ( )10 0.9,0.99,0.999, 9,9.9,9.99,⋅ = but it is impossible to isolate one 9 from infinitely many terms.

3. Conclusions As a result we can state that an infinite digit sequence like 1 2 30.d d d , abbreviating an infinite sequence of

partial sums of decimal fractions, also called an infinite series 1 10 10

nk nk n

k nn

d d= ∈∈

≡ ∑ ∑

, is not a number (unless

eventually becoming constant). 3.1415 for example is an abbreviation of the sequence of rational partial sums converging to π. This sequence is purely rational although we cannot find a fraction 3.1415m n = with a common denominator covering all terms of the sequence. This disadvantage however is shared by sequences like ( )10 n

n

−

∈ too. We cannot find a fraction with a common denominator covering all terms of the sequences

all of which are rational with no doubt. A periodic decimal fraction has as its limit a rational number. A non-periodic decimal fraction has as its limit

an irrational number. But it is not this number. In case of periodic decimal fractions it is possible, by changing the basis, to obtain a terminating digit sequence. Irrational numbers have no decimal expansion, no representa-tion by digits or bits, not even by infinitely many. They are incommensurable with every rational-measure ex-panded by digits or bits. An irrational number requires a generating formula F in order to calculate every digit of the infinite digit sequence S and in addition to calculate the limit. The formula F may be interpreted as the number as well as the limit. It may be involved or as simple as “0.111…” which is a finite formula (consisting of eight symbols) allowing to obtain every digit of the infinite sequence converging to 1/9.

The implication F S⇒ cannot be reversed because without F the sequence S cannot be obtained in the completeness required, i.e., including all its terms such that none is missing.

4. Consequence The mathematical facts discussed above also apply to all sequences of digits or bits appearing in the folklore version of Cantor’s diagonal argument [3] or in the binary tree argument [4]. Sequences of digits or bits are never representing irrational numbers, let alone transcendental numbers. Therefore Cantor’s diagonal argument, as well as the binary tree argument, does not concern the cardinality of the set of irrational numbers.

References [1] Mueckenheim, W. (2011) Mathematik für die ersten Semester. 3rd Edition, Oldenbourg Verlag GmbH, Muenchen, 193.

http://www.amazon.de/Mathematik-f%C3%BCr-die-ersten-Semester/dp/348670821X/ref=sr_1_2?s=books&ie=UTF8&qid=1400566108&sr=1-2&keywords=Mathematik+f%C3%BCr+die+ersten+Semester

[2] Cantor, G. (1889) Bemerkungen mit Bezug auf den Aufsatz: Zur Weierstraß-Cantorschen Theorie der Irrationalzahlen. Mathematische Annalen, 33, 476. http://dx.doi.org/10.1007/BF01443973

[3] Cantor, G. (1891) Über eine elementare Frage der Mannigfaltigkeitslehre. Jahresbericht der Deutschen Mathematiker- Vereinigung, 1, 75-78.

[4] Mueckenheim, W. (2008) The Infinite in Sciences and Arts. In: Sriraman, B., Michelsen, C., Beckmann, A. and Fre-iman, V., Eds., Proceedings of the 2nd International Symposium of Mathematics and Its Connections to the Arts and Sciences (MACAS2), Centre for Science and Mathematics Education, University of Southern Denmark, Odense, 265- 272. http://static.sdu.dk/mediafiles//Files/Om_SDU/Centre/C_NAMADI/Skriftserie/MACAS_samlet.pdf